No X Points on a Y: A Class of Discrete Geometry Problems

Richard Voepel, Rutgers University

Abstract: First introduced in 1917 by Henry Dudeney, the No-Three-In-Line problem asks for the maximum number of points that can be placed in an N by N grid such that no three are collinear. While there have been results concerning lower bounds for this number, non-trivial upper bounds remain largely conjectural. But this is not the only problem of this form to receive attention; one may consider generalizations to higher dimensions, asking for no three points to be collinear in an N by N by N grid, or for no four points to be coplanar. We present select results for these problems, and propose further cases to study.

Copyright 2019, Rutgers, The State University of New Jersey. All rights reserved.
Rutgers is an equal access/equal opportunity institution. Individuals with disabilities are encouraged to direct suggestions, comments, or complaints concerning any accessibility issues with Rutgers web sites to: accessibility@rutgers.edu or complete the Report Accessibility Barrier or Provide Feedback Form.